Question

use mathematical induction prove that 2+3.2+4.2^2+.... upto n terms=n 2^n for all n€N​

Answer 1

Answer:

$2+3.2+4.2^2+5.2^3+..... \text{upto n terms}=n2^n$    ........ (1)

For n = 1

$2=2$

Thus (1) is true for n = 1

Let us assume that

for n = k

$2+3.2+4.2^2+5.2^3+..... \text{upto n terms}=k2^k$ is true

i.e. $2+3.2+4.2^2+5.2^3+..... +(1+k).2^{k-1}}=k2^k$ is true

Adding (2+k).2^k on both sides

$2+3.2+4.2^2+5.2^3+..... +(1+k).2^{k-1}+(2+k).2^k=k2^k+(2+k).2^k$

$\implies 2+3.2+4.2^2+5.2^3+..... +(1+k).2^{k-1}+(2+k).2^k=2k2^k+2.2^k$

$\implies 2+3.2+4.2^2+5.2^3+..... +(1+k).2^{k-1}+(2+k).2^k=k2^{k+1}+2^{k+1}$

$\implies 2+3.2+4.2^2+5.2^3+..... +(1+k).2^{k-1}+(1+k+1).2^k=(k+1)2^{k+1}$

Thus the relation holds true for n = k ⇒ it holds true for n = k+1

Thus the relation holds true for all natural numbers n         (Proved)